5 THE TIETZE-URYSOHN EXTENSION THEOREM 89

which shows that \z — z'\ = d(z, z') is a distance defined on C = R x R,
which is uniformly equivalent to the distance considered in Section 3.20.
The balls for that distance are called discs. Any complex number z ^ 0
can be written in one and only one way as a product r£, with r > 0 and
|C| = 1, namely by taking r = \z\ and £ = z/\z\.

PROBLEM

Let /be a continuous mapping of C into itself such that /(z -f z') =/(z) +/(zO and
/(zzO =/(z)/(z'). Show that either /(z) = 0 for every z e C, or / is one of the mappings
z -»- z, z -> z (use (4.1.3)). (It can be proved, using the axiom of choice, that there aremjec-
tive, nonsurjective and noncontinuous mappings / of C into itself, such that /(z + z') =
f(z) +/OO and/(zz') = /(z)/(z'). Compare to the problem in Section 4.2.)

5. THE TIETZE-URYSOHN EXTENSION THEOREM

(4.5.1) (Tietze-Urysohn extension theorem) Let "E be a metric space,
A a closed subset of E, f a continuous bounded mapping of A into R. Then
there exists a continuous mapping gof^E into R which coincides with f in A
and is such that

sup g(x) = sup/(jO, inf g(x) = inf/Cy).

jceA yeA jeeE yeA

We may suppose that inf f(y) = 1 , sup f(y) = 2 by replacing eventually

y e A y e A

/by a mapping y ->• a/W + /?, a 7^ 0 (the case in which/is constant is trivial).

Define g(x) as equal tof(x) for x e A, and given by the formula

yeA

for x e E - A. From the inequalities 1 </(.y) < 2 for y e A and the definition
of d(x, A), it follows that 1 < g(x) ^ 2 for jc 6 E - A. We need therefore
only prove the continuity of g at every point x e E. If x e A, the continuity
follows from the assumption on /. In the open set E — A, we can write
g(x) = h(x)/d(x, A) with h(x) = inf (f(y) d(x, y)), and as d(x, A) is continuous

yeA

and 7*0 (by (3.8.9) and (3.11.8)), all we have to prove then (by (4.1.2) and
(4.1.4)) is that h is continuous at every xeE — A. Let r = d(x, A); for
d(x, x) ^e < r, we have d(x, y) *$ d(x', y) + e, hence /z(x) ^ /z(x') -f 2e
(since /(;;) < 2), and similarly h(xf) < /z(x) -f- 2e, which proves the continuity
of h. Finally, let us suppose x is a frontier point of A; given e > 0, let r > 0on, since